Finite Group Databasebeta

← #406775

Subgroups — #406775

Order 512.

1266 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20844C211yesmaximal, characteristic
2#20834C211yesmaximal, characteristic
3#20700C211yesmaximal, characteristic
4C2×#1821C211yesmaximal, characteristic
5C2×#1810C211yesmaximal, characteristic
6C22×#340C211yesmaximal, characteristic
7#11303C212yesmaximal, complemented
8#11249C212yesmaximal, complemented
9#11242C212yesmaximal, complemented
10#11216C212yesmaximal, complemented
11#10727C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#108C2212yes—
16C2×#428C2212yes—
17C2×#424C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#407C2211yescharacteristic
20C2×#407C2211yescharacteristic
21C2×#407C2211yescharacteristic
22#2280C2211yescharacteristic
23#1853—21no—
24#1853—21no—
25#1852—21no—
26#1852—21no—
27#1850—22no—
28#1850—22no—
29#1824—21no—
30#1824—21no—
31#1823—21no—
32#1823—21no—
33#1821—24no—
34#1810—24no—
35C2×#343C2212yes—
36C2×#343C2212yes—
37C2×#343C2212yes—
38C2×#343C2212yes—
39C2×#341C2212yes—
40#1736C2212yes—
41#1736C2212yes—
42#1736C2212yes—
43C2×#340—22no—
44C2×#340—22no—
45C2×#340C2212yes—
46C2×#340C2212yes—
47#1729C2211yescharacteristic
48#1729C2211yescharacteristic
49C24×C4—21no—
50C24×C4—21no—
51C24×C4C2311yescharacteristic
52C22×M16C2311yescharacteristic
53C22×M16C2312yes—
54C22×M16C2311yescharacteristic
55C22×M16C2312yes—
56C23×C8—22no—
57C23×C8—22no—
58C23×C8—24no—
59C23×C8—42no—
60C23×C8D412yes—
61C23×C8D412yes—
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C2×#117C2311yescharacteristic
65C2×#117C2311yescharacteristic
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—24no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—42no—
81C22×C4⋊C4Q814yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C2×#108—24no—
85C2×#108—24no—
86C2×#108—42no—
87#428—24no—
88#428—42no—
89C2×#107—22no—
90C2×#106—22no—
91#424—24no—
92#424—42no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95#407—22no—
96#407—22no—
97#407—22no—
98#407—22no—
99#407—22no—
100#407D414yes—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107#381—21no—
108#380D411yescharacteristic
109#380D411yescharacteristic
110#380D411yescharacteristic
111#380D411yescharacteristic
112#379—21no—
113C4×C22⋊C4—21no—
114C4×C22⋊C4—21no—
115#343—24no—
116#343—24no—
117#343—24no—
118#343—42no—
119#341—24no—
120#340—24no—
121#340—24no—
122#340—22no—
123#340—22no—
124#340—22no—
125#340—22no—
126#340—22no—
127#340—22no—
128#336—22no—
129#336—22no—
130C25#4111yescharacteristic
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—24no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—44no—
139C23×C4—41no—
140C23×C4—41no—
141C23×C4—41no—
142C23×C4—41no—
143C23×C4—42no—
144C23×C4—82no—
145C23×C4—82no—
146C23×C4—82no—
147C23×C4—82no—
148C23×C4C2×D412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×Q812yes—
154C23×C4#4111yescharacteristic
155C23×C4#4112yes—
156C23×C4#4112yes—
157C23×C4C2411yescharacteristic
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—21no—
165C2×M16—21no—
166C2×M16—22no—
167C2×M16—22no—
168C2×M16—22no—
169C2×M16—21no—
170C2×M16—21no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—22no—
175C2×M16—22no—
176C2×M16—22no—
177C2×M16C2×D414yes—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—24no—
182C22×C8—24no—
183C22×C8—21no—
184C22×C8—22no—
185C22×C8—21no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—24no—
189C22×C8—24no—
190C22×C8—21no—
191C22×C8—22no—
192C22×C8—21no—
193C22×C8—42no—
194C22×C8—42no—
195C22×C8—48no—
196C22×C8—48no—
197C22×C8C2×D412yes—
198C22×C8C2×D412yes—
199C22×C8#4114yes—
200#117—22no—
201#117—22no—
202#117—22no—
203#117—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×Q814yes—
240C2×C4⋊C4#4114yes—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C42—21no—
248C2×C42—21no—
249#108—48no—
250#107—44no—
251#106—44no—
252C4⋊C8—44no—
253C4⋊C8—44no—
254C22⋊C8—24no—
255C22⋊C8—24no—
256C22⋊C8—42no—
257C22⋊C8—42no—
258C22⋊C8—42no—
259C22⋊C8—42no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24#12111yescharacteristic
271C24#12111yescharacteristic
272C24C22⋊Q812yes—
273C24#12612yes—
274C24C2×#4111yescharacteristic
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—816no—
312C22×C4—88no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4—88no—
322C22×C4—82no—
323C22×C4—82no—
324C22×C4—82no—
325C22×C4—82no—
326C22×C4#12112yes—
327C22×C4#12112yes—
328C22×C4#12112yes—
329C22×C4#12112yes—
330C22×C4#12111yescharacteristic
331C22×C4#12111yescharacteristic
332C22×C4C22⋊Q812yes—
333C22×C4C22⋊Q812yes—
334C22×C4C22⋊Q812yes—
335C22×C4C22⋊Q812yes—
336C22×C4C22⋊Q812yes—
337C22×C4#12612yes—
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C22×Q811yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4C2×#4111yescharacteristic
344C22×C4C2×#4112yes—
345C22×C4C2×#4112yes—
346C22×C4H(5,𝔽2)12yes—
347C22×C42−1+412yes—
348M16—24no—
349M16—24no—
350M16—42no—
351M16—42no—
352M16—42no—
353M16—42no—
354M16—42no—
355M16—42no—
356M16—42no—
357M16—42no—
358M16—42no—
359M16—42no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—48no—
369C2×C8—48no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—42no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8#12314yes—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—22no—
388C4⋊C4—24no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—44no—
399C4⋊C4—44no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C42—41no—
407C42—41no—
408C42—41no—
409C42—41no—
410C23—28no—
411C23—28no—
412C23—28no—
413C23—28no—
414C23—24no—
415C23—24no—
416C23—24no—
417C23—24no—
418C23—24no—
419C23—21no—
420C23—21no—
421C23—24no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23#39612yes—
433C23#39612yes—
434C23#47112yes—
435C23#47112yes—
436C23#47212yes—
437C23#47312yes—
438C23C2×#12111yescharacteristic
439C23C2×C22⋊Q811yescharacteristic
440C23C2×#12611yescharacteristic
441C23#55112yes—
442C23#55212yes—
443C2×C4—24no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—24no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—44no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—44no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—84no—
476C2×C4—88no—
477C2×C4—84no—
478C2×C4—88no—
479C2×C4—88no—
480C2×C4—88no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—84no—
486C2×C4—84no—
487C2×C4#39612yes—
488C2×C4#39612yes—
489C2×C4C2×#12111yescharacteristic
490C2×C4C2×#12111yescharacteristic
491C2×C4C2×C22⋊Q811yescharacteristic
492C2×C4C2×C22⋊Q811yescharacteristic
493C2×C4C2×#12311yescharacteristic
494C2×C4#52511yescharacteristic
495C2×C4#52912yes—
496C2×C4#53312yes—
497C2×C4#53411yescharacteristic
498C2×C4#53611yescharacteristic
499C2×C4#53912yes—
500C2×C4#54412yes—
501C2×C4#54612yes—
502C2×C4#54612yes—
503C2×C4#54712yes—
504C2×C4#54712yes—
505C2×C4#55112yes—
506C2×C4#55212yes—
507C8—24no—
508C8—42no—
509C8—44no—
510C8—81no—
511C8—81no—
512C8—82no—
513C22—216no—
514C22—216no—
515C22—22no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—21no—
522C22—21no—
523C22—22no—
524C22—21no—
525C22—21no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22#206314yes—
536C22#206414yes—
537C22C2×#39611yescharacteristic
538C22#243812yes—
539C22#248611yescharacteristic
540C22#249312yes—
541C22#253411yescharacteristic
542C22C2×#47111yescharacteristic
543C22C2×#47111yescharacteristic
544C22C2×#47211yescharacteristic
545C22C2×#47311yescharacteristic
546C4—22no—
547C4—22no—
548C4—81no—
549C4—81no—
550C4—84no—
551C4—81no—
552C4—84no—
553C4—81no—
554C4#241211yescharacteristic
555C4#241311yescharacteristic
556C4#242512yes—
557C4#243812yes—
558C4#262711yescharacteristic
559C4#285511yescharacteristic
560C2—24no—
561C2—24no—
562C2—22no—
563C2—22no—
564C2#1493511yescharacteristic
565C2C2×#206311yescharacteristic
566C2C2×#206411yescharacteristic
567C2#2767211yescharacteristic
568C2#2767511yescharacteristic
569C2#2813411yescharacteristic
570C2#2813511yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
7. #11303C2semidirect
8. #11249C2semidirect
9. #11242C2semidirect
10. #11216C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#208442
#208342
#207002
C2×#18212
C2×#18102
C22×#3402
#113032
#112492
#112422
#112162
#107272